On the total number of parts functions associated with ranks of partitions modulo 5 and 7

Recently, Andrews proved congruences for the total number of parts functions associated with ranks of partitions modulo 5 and 7. In this paper, applying the method of Atkin and Swinnerton-Dyer, we establish identities for these functions from which Andrews’ congruences modulo 5 follow immediately. In particular, the generating functions for total number parts functions associated with ranks of partitions modulo 7 turn out to be theta-functions.